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Open Science Journal – October 2019 1
RESEARCH ARTICLE
The Calibration of Hailpads upon the Greek
National Suppression Program, using the
Classical and Inverse Regression Methods
Kyriakos G. Tsitouridis1*
1Meteorological Applications Centre, Thessaloniki, Greece
*Corresponding author: Kyriakos G. Tsitouridis: k.tsitouridis@elga.gr
Abstract:
Citation: Tsitouridis K.G. (2019) The Calibration of Hailpads upon the Greek National Suppression Program, using the Classical and Inverse Regression Methods. Open Science Journal 4(1) Received: 28th October 2018 Accepted: 18th September 2019 Published: 16th October 2019 Copyright: © 2019 This is an open access article under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: The author(s) received no specific funding for this work Competing Interests: The author have declared that no competing interests exists.
The hailpad, constructed from a plate of Styrofoam, is a simple
instrument for recording hailfall. In addition to simply recording
the hailfall, calibration of the instrument is required to obtain
quantitative measurements of the hail. The calibration is a
process leading to a calibration equation, a polynomial
establishing a relationship between the diameter of a hailstone
and the dent the hailstone is left on the surface of the hailpad. A
hailpad network, consisted of 154 instruments, has established in
Greece, in the context of the Greek National Hail Suppression
Program operating for the protection of the agricultural
cultivations from hail damage. For the calibration of the haipads
of the network the well known “Energy Matching technique” has
adopted and the Inverse Regression method is applied from the
beginning of the operation of the network on 1984, for the
obtainment of the calibration equation. In the present study,
along with the Inverse Regression method hitherto applied, the
Classical Regression method is examined and presented and, for
the first time, inferential statistics are also introduced in order to
establish a more stringent statistical procedure for the
calibration of the hailpads. After the theoretical analysis of the
two methods, the data from a calibration experiment were
analyzed, calibration models obtained using both methods of
regression, hail diameters were predicted with the two models
and the results compared to each other. The comparison of the
predictions with the two models shows that the very small
differences that appear can be ignored, taking into account and
the natural variability of the properties of the hail, so classical
regression does not have any advantage over reverse regression.
It is therefore shown that reverse regression, which is also used
in other networks internationally, is quite satisfactory and as
Open Science Journal Research Article
Open Science Journal – October 2019 2
Keywords: Hailpad, calibration, regression, classical regression, inverse
regression, prediction bands
Introduction
The hailpad is a simple yet effective meteorological instrument for the study
of the hail falls. The instrument introduced in 1959 [1] and has since been used by
many researchers, for the study of the characteristics and climatology of hail and
for the evaluation of hail suppression programs [2-6].
The hailpad was initially introduced in the Greek National Hail Suppression
Program – GNHSP - during the period 1984 – 1988 when a hailpad network was
established in the context of the evaluation experiment of the program based on a
crossover design. After the end of the evaluation experiment, the hail suppression
program continued as purely operational for the protection of the agricultural
cultivations from hail damage and the Meteorological Applications Centre, a
branch of the Hellenic Agricultural Insurance Organization-ELGA, is responsible
for the operation of the Program. For the daily evaluation of the Program the
hailpad network consisted of 154 instruments was maintained in operation till
now.
There are few variations of the instrument [5, 7]. The variation of the
instrument used in the GNHSP, consists of a pad of Styrofoam mounted
horizontally in a case at the tope of a pole, about 1.5m above the ground, so that
the upper surface exposed to the hail has dimensions of 27cmX27cm. The
Styrofoam variation with the smooth skin is used, which is painted with a water
mixture of white wall painting emulsion for protection against the sun radiation.
The response of the instrument is a dent formed in the surface of the pad,
after the impact of a hail stone. From the dent that causes the hail stone on the
surface of the hailpad, the researcher has to draw conclusions about the
characteristics of the hail, so the calibration of the instrument is an essential
stage for the measurement of certain characteristics of the hailstone. The
calibration is a group of procedures that lead to the establishment of a
mathematical relationship between the calibration standards used – calibration
steel spheres - and the responses of the instrument. This mathematical
relationship is then used to transform the responses of the instrument - the
minimum diameters of the dents left on the surface of the pad - to estimates of
the diameters of the hail stones responsible for the formation of the dents. The
calibration of the hailpads in the GNHSP is performed applying the “Energy-
Matching - EM” technique, originated with Schleusener and Jennings [1].
The practice hitherto upon the GNHSP is the obtainment of the calibration
equation using the Inverse Regression method as shown below, without
considering whether the method used is the most appropriate and whether the
simple as its implementation, it is proposed as the main
calibration method. Inferential statistics is suggested to be
applied and prediction bands to be calculated in each future
calibration experiment, giving greater validity to the results. In
addition to this, the classical regression method can be used for
continuous comparison of the results between the two methods.
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calibration data meets the requirements of the regression theory. The purpose of
this work is to seek answers to these questions, to examine the alternative
Classical Regretion method and to set a more stringent statistical procedure for
the calibration of the hailpads.
For the obtainment of the calibration equation two different methods of linear
regression are presented, the hitherto used Inverse Regression method and the
Classical Regression method as well. Finally the data of a calibration experiment
are analyzed with the two methods and prediction bands for new observations are
produced.
The remainder of the work is organized as follows:
In section 2, the operation of the hailpad network is presented, along with the
theoretical basis, the experimental procedures and the statistical methods for the
analysis of the experimental data. In section 3 an application example is
presented, analyzing the data of a calibration experiment, using both methods of
regression, prediction bands are derived for both and finally the results are
compared, closing with conclusion and recommendations.
Materials and methods
The operation of the hailpad network
Upon the GNHSP, a network of hailpad stations is in operation from 1984 till
now, consisted of 154 stations from 2008 till now, in an area of 2,200 square km,
with a mean of 3.8X3.8km2 area corresponding to each hailpad. During the hail
period from March to September each year, the network is under continuous
service. Every hailpad is replaced by a new one in two different instances, a) if it
is affected by a storm b) if it is exposed to the sun radiation for more than 30 to
35 days, time period during which it keeps it’s initial properties [5].
The identification of the hailpads affected by a storm is carried out using the
Thunderstorm Identification, Tracking, Analysis and Nowcasting – TITAN -
system [8], a software ingesting radar data from the EEC C-band meteorological
radar installed at Filyron, a village about 20 km north of Macedonia Airport of
Thessaloniki, Greece. The hailpads affected by storms in a day, are replaced in
the next day as soon as possible. A common problem of a hailpad network is the
possibility a hailpad to be affected by different storms in a day, a problem faced
effectively by the use of the TITAN software and the analysis of the storms.
The hit hailpads are painted with black ink by means of a printmaking
cylinder, in order the boundaries of the dents to become clear, and then scanned
so that the hailpad surface to be converted to a digital image, ready for the
analysis. The analysis is performed using an application built on the
ImageProPlus software.
The Calibration of the Hailpads - Theory and Practices
Two groups of assumptions are accepted for the calibration of the hailpads,
the first group is related to the physical properties of the hail stone and the
Styrofoam and the second one is related to the statistical theory behind the
regression method. For the purposes of this paper, both groups of the
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assumptions are briefly quoted. For a deeper study of the assumptions the reader
may have recourse to the cited literature.
For the calibration of the hailpads and the data reduction certain assumptions
are accepted as stated in [1, 3, 5]. Those assumptions accepted upon the GNHSP
too, are briefly the following: a. The hailstone has a spherical shape, b. The
hailstones that impact with the ground are rigid, c. The density of all hailstones
is ρh=0.89g/cm3, d. The drag coefficient of the hailstones is Cd=0.6, e. A
hailstone falls vertically onto a hailpad, with its terminal velocity and f. The
hailstones hit the hailpad once.
These assumptions are based on a summary of the findings of the research and
in no case represent any individual hail fall, as hail has a great variability in
nature [9], but they are a good basis for assessing the characteristics of the hail
falls. The alternative, more accurate, methods are based on very expensive
equipment to collect the real hailstones [10].
The purpose of the calibration of the hailpad is to establish a mathematical
relation between the minor axis of a dent and certain properties of the hail stone
making the dent [5]. As already mentioned in the Introduction, in the GNHSP
the “Energy-Matching - EM” calibration technique has been adopted, originated
with Schleusener and Jennings [1].
The EM technique does depend on the assumption that “spheres of equal
diameters create dents of equal minor axes when the spheres have equal impact
kinetic energy”. As reported on [5] “Lozowski et al (1978a) have shown
experimentally that this assumption holds for spheres of ice (ρ=0.9 g/cm3), glass
(ρ=2.5 g/cm3) and steel (ρ=7.8 g/cm3). Experiments carried out in the National
Hail Research Experiment - NHRE with polypropylene spheres (ρ=0.9 g/cm3)
and teflon spheres (ρ=2.2 g/cm3) confirmed this result”, p1305. The theory of
dent formation first proposed by Lozowski et al [3] and developed further by
Long et al., [5, 7], also supports this assumption. Not any similar experiment
carried out in the GNHSP, but the dent formation theory referred in the
mentioned literature has been adopted.
For the calibration of the hailpads, a steel simulation sphere is dropped from
the proper height, so that the kinetic energy of the simulation sphere at impact
with the hailpad is equal to the kinetic energy of a hailstone with the same
diameter, falling with its terminal velocity.
In Long et al., [5, 7] the mathematical expression (1) obtained for the height
from which a simulation sphere must be dropped in still air onto a hailpad so
that the impact kinetic energy of the sphere equals that of a spherical hailstone of
the same diameter, falling with its terminal velocity.
s h a2 1 ds h a1s
a2 ds s a1 2 dh s a2
2 g C ( )H ln 1 D
3 C g C ( )
• • • • • −= − • − •
• • • • • − (1)
In (1) ρs, and ρh are the densities of steel and hail respectively, Cds, Cdh are
the drag coefficients (dimensionless) of the steel and the hail respectively and D is
the diameter of the simulation sphere, ρa1 is the density of air and g1 is the
acceleration of gravity near the field where the data are collected, while ρa2 is the
density of air and g2 is the acceleration of gravity in the calibration laboratory.
Taking into account that the laboratory of the GNHSP is close to the hailpad
network and there is not significant difference in the longitude, latitude and
altitude between the hailpad locations and the laboratory location, and assuming
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still air, it is accepted that the air densities are the same for both locations:
ρa1=ρa2=ρ. Furthermore the calculations shown that the involved gravitational
accelerations in the field and the laboratory are equal: g1=g2=g. So from the
equation (1) the following simplified expression (2) is obtained:
s h ds h as
a ds s dh s a
2 C ( )H ln 1 D
3 C C ( )
• • • −= − • − •
• • • − (2)
Substituting in the above equation (2) the values of the involved constant
parameters Density of air: ρa =0.00119g/cm3=1.19kg/m3, Density of (steel)
sphere: ρs=7.78g/cm3=7780kg/m3, Density of hail: ρh=0.89g/cm3=890kg/m3,
Drag coefficient of the steel spheres: Cds=0.45 (dimensionless), Drag coefficient of
hail: Cdh=0.60 (dimensionless), the simplified equation (3) results:
sH 95.418764 D= • (3)
Substituting in the above simplified equation the diameters of the twelve (12)
simulation spheres used in the GNHSP, the following twelve pairs of values of the
Table 1 are obtained, where Ds is the diameter of the simulation sphere and Hs is
the drop height, both expressed in mm.
Table 1. Drop heights of the simulation spheres
No Ds (mm) Hs (mm) No Ds (mm) Hs (mm) No Ds (mm) Hs (mm)
1 6.35 606 5 12.70 1212 9 22.23 2121
2 7.94 758 6 13.49 1287 10 25.40 2424
3 9.53 909 7 14.29 1364 11 31.75 3030
4 11.11 1060 8 18.26 1742 12 34.93 3333
The minor axes of the dents left on the surface of the hailpad by the steel
spheres are measured using the ImageProPlus software, and the data are used to
develop a linear model relating the diameter of the sphere to the minor axis of the
dent, by means of the least squares method.
Till the year 2010 a calibration experiment carried out at the beginning of the
hail period for the total consignment of the hailpad material. From the year 2011
a calibration experiment is carried out for every packet of Styrofoam consisted of
14 plates with dimensions of 60cmX220cm, to minimize the errors related to the
painting and inking procedures and the variability of the properties of the
material. It could be better to apply the individual calibration method which is
promising smaller errors [12] but the transition to this method is very difficult for
practical and economical reasons, so the packet calibration is a good alternative
as there is no need for the replacement of the existing infrastructure.
Beyond the above change, some improvements have been made between 2007
and 2009 with regard to calibration procedures. Particularly, a hailpad laboratory
has been developed and equipped with a calibration device, a new inking cylinder
with dimensions appropriate to cover a hailpad with black ink in one revolution
and an A3 calibrated scanner. In addition, the analysis of the hailpads and the
data reduction is performed using an application built on the ImageProPlus
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software. With these improvements, the error sources were drastically reduced
and the run time of the calibration experiments is limited. Furthermore a hailpad
technician has been trained to carry out all the necessary work, the painting,
inking and scanning included, in order to greatly reduce the human error.
Under the new laboratory conditions and practices, which guarantee fewer and
smaller measurement errors, there is the necessary background for introducing
more stringent statistical analyses of the calibration data, enriched with
statistical inference procedures.
Summary of the statistical methods for the calibration.
According to the research and the experience, in the case of hailpads there is a
causal relationship of the form Y=f(X), between the diameter X of a hailstone
and the diameter Y of the dent left by the hailstone on the surface of the hailpad.
The most common relationship used is the linear one of the form Y=β0+β1X,
while two or higher degree polynomials are proposed [11]. The coefficients β0 and
β1 are estimated by regression procedures as shown below in this paragraph.
Due to the possible packet-to- packet variability of the hailpad material, a
calibration experiment is carried out for every new packet of Styrofoam plates
from the year 2011. Each one of the twelve (12) simulation steel spheres of the
Table 1 is released ten (10) times from the proper height to hit the surface of
sample hailpads collected from a new packet consisted of 14 plates of Styrofoam.
After the end of the experiment, every hit sample hailpad is inked and scanned.
Using the ImageProPlus software, the minimum diameters of the dents are
measured and a table of 120 pairs of values (Xi, Yi) is formed, where Xi is the
diameter Ds of the simulation steel sphere and Yi is the minor axis Dd of the dent
on the surface of the hailpad.
In the case of the hail, naturally, the diameter of the hailstone (and the
diameter - Ds - of the steel simulation sphere as well) is the independent variable
X, as the hailstone is the cause of the dent formed on the surface of the hailpad,
and the diameter of the dent - Dd -is the dependent or response variable Y.
As stated in Weiss, Neel A, 2012, p. 670, [13] certain assumptions (conditions)
are accepted for the regression model:
“1. There are constants β0 and β1 such that, for each value X of the predictor variable, the conditional mean of the response variable is β0+β1X.
2. The conditional standard deviations of the response variable are the same for all values of the predictor variable.
3. For each value of the predictor variable, the conditional distribution of the response variable is a normal distribution.
4. The observations of the response variable are independent of one another”.
In the calibration of the hailpads, the diameter and the density of the
simulation spheres are involved in the obtainment of equation (1) and they are
measured with a very small error which can be ignored, while the measurement of
the diameter of the dent formed on the surface of the Styrofoam is subject to
errors, because of various parameters such as the differences of the properties of
Styrofoam mass from point to point, the different thickness of the ink film, the
tiny anomalies of the Styrofoam surface, the errors associated with the analysis
software and so on. Also, the above assumptions 2 and 3 are supposed to be
valid.
In the case of the calibration of hailpads, during the experimental procedure it
is ensured that the replicate drops of a simulation sphere are absolutely
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independent of one another and care is taken to ensure that each dent is formed
away from the previous one, so the assumption 4 is always valid.
A good approach to building a regression model so as not to violate the basic
assumptions mentioned above is to consider the diameter of the steel as the
independent variable and the diameter of the dent as the dependent variable as
this happens naturally. This is the method of Classical Regression. However, the
Inverse Regression method is widely used not only for the calibration of the
hailpads but generally in the calibration of measuring devices. According to this
method, the diameter of the dent is considered as the independent variable and
the diameter of the steel ball is considered as the dependent variable.
The two methods examined and compared by many researchers. Krutchkoff,
R.G.,[14] compared the two methods of linear calibration by Mode Carlo methods
and concluded that the Inverse Regression is slightly better than the Classical
one. Shucla, G.K.,[15] concluded that both methods have advantages depended on
the sample size and suggested the classical estimators for general purposes and
large sample sizes. Parker et al [16] made a similar comparison of the two
methods and proposed prediction intervals for both.
In the case of the Inverse Regression for the calibration of the hailpads, the
value of the minor axis of the dent, which is used as predictor, depends on
various parameters, measured with not negligible errors, as mentioned above. So
this method is expected to “violate the simple linear regression assumption that
the predictor is measured with negligible error”,[16].
In this paper, both, the Classical Regression and the Inverse Regression are
presented. After the theoretical presentation of the two methods, the same data
set from a hailpad calibration experiment is used to build a linear model and
construct prediction intervals in the case of the Classical Regression and in the
case of Inverse Regression.
In both methods, the same regression and statistical inference theory apply. In
Montgomery et al [17], (Chapter 11), there is a comprehensive presentation of the
Simple Linear Regression method, which is briefly presented in the following.
Let X to be the independent (predictor) variable and Y the dependent
(response) variable. The values Xi of X are all measured with negligible error
while the values of Yi of Y are measured with random errors. Assuming that a
simple linear model is appropriate, the true model is:
i 0 1 i iY X= + + (4)
In the model (4) β0 is the Y-intercept, β1 is the slope of the line, and the εi’s
are random errors. The random errors εi assumed to be independent and normally
distributed with a mean of zero (0) and a variance of σ2, εi~N(0,σ2).
The regression function (5) based on n observations is obtained, using the
least squares method.
0 1Y b b X
= + (5)
In the equation (5) b0 and b1 are unbiased point estimators of β0 and β1
respectively, given by the equations (6) and (7) [17].
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( )( )n
i ii 1 XY
1 n2
XXi
i 1 1
X X Y YS
bS(X X)
=
=
− −
= =
−
(6)
0 1b Y b X= − (7)
After the equation (5) is obtained, a hypothesis testing is performed to
examine if β 1≠0 and β 0≠0, using the test statistic i i it* (b ) s b= − ,
which follows the t-distribution, because the population statistics are unknown
and only sample statistics are available.
For (1-α)100% confidence interval the null hypothesis H0: β1=0 versus the
alternative Ha: β 1≠0 is examined. Setting β 1=0 the t* statistic becomes:
t*=b1/s{b1}. A two tailed test is performed and the null hypothesis H0 is rejected
and the alternative Ha is accepted if |t*|>t(1-α /2;n-2), where t(1-α /2;n-2) is the
critical value of the t-distribution of the two tailed test for α level of significance
and n-2 degrees of freedom. The real meaning of β1≠0 is that the slope β1 is
significantly different from zero, or otherwise for an increase of the independent
variable by 1.0, there is an increase in the dependent variable by β1.
Following a similar procedure, a hypothesis test is performed about the
intercept β0. For (1-α)100% confidence interval the null hypothesis H0: β0=0
versus the alternative Ha: β 0≠0 is examined. Setting β 0=0 the t* statistic
becomes: t*=b0/s{b0}. The null hypothesis H0 is rejected and the alternative Ha
is accepted if |t*|>t(1-α/2;n-2).
Given the point estimators b1 of β1 and b0 of β0, (1-α)100% confidence limits
for β1 and β0 are:
1 1(1 ;n 2)
2
b t s b− −
• and 0 0(1 ;n 2)
2
b t s b− −
• (8)
From the fitted model (5), for a given value x0 of the predictor variable X,
the mean estimated value 0y
of the response variable is
0 1 00y b b x
= + (9)
A (1-α)100% confidence interval for 0y
is given by the expression (10).
2 22 20 0
0 0 0(1 ;n 2) (1 ;n 2)
xx xx2 2
(x X) (x X)1 1y t s y y t s
n S n S
− − − −
− −− • + + • +
(10)
The above confidence interval is minimized at0x X= and is widening as
0| x X |− increases.
Another way to judge the adequacy of the regression model is to calculate the
coefficient of determination N N
2 2 2
iR T i
i 1 i 1
SS SS (Y Y) (Y Y)R
= =
= = − − (11)
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OrN N
2 2 2iR T E T i i
i 1 i 1
SS SS 1 SS SS 1 (Y Y ) (Y Y)R
= =
= = − = − − − (12)
As closer to 1 is the value of the R2 as stronger the correlation between the two
parameters is.
The Classical Regression method.
In the case of the Classical Regression method also known as Inverse
Prediction, when a new observation 0Y of a dent becomes available, the point
estimation of the level X0, the diameter of a hailstone, that gave rise to this new
observation is made by inverting the model (5), so that given Y0 and 1b 0 , a
point estimator of X0 is: 0 0
0
1
Y bX
b
−= (13)
0 0 0(1 ;n 2) (1 ;n 2)
2 2
X t s predX X X t s predX
− − − −
− • + • (14)
In (23) 2s predX s predX= and 22
20 xx2
1
ss predX 1 1/ n X X / S
b
= • + + −
(15)
and the (1 ;n 2)
2
t − −
is the critical value of the t-distribution of the two tailed
test for α level of significance and n-2 degrees of freedom.
As can be seen in (15), the above prediction interval of 0X
is minimized at
0X X
= and is widening as 0| X X |
− increases.
The Inverse Regression method.
In the case of the Inverse Regression method, as stated by [16], [18] (pp 58-
59), the variable X, the diameter of the simulation sphere, is treated as the
response (dependent) variable and the Y, the minimum diameter of the created
dent, is treated as the predictor (independent) variable. Then the model (4) can
be written in the form (16).
'
i 0 1 iX Y = + + (16)
In the model (16) γ0 is the X-intercept (ordinate), γ1 is the slope of the line and '
i are random errors. The random errors '
i assumed to be independent and
identically distributed as normal with a mean of zero (0) and a variance of σ2, '
i ~N(0,σ2).
The regression function (17) based on n observations is obtained, using the least
squares method.
0 1X c c Y
= + (17)
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In the equation (17) c0 and c1 are unbiased point estimators of γ0 and γ1
respectively, given by the equations (18) and (19):
( )( )n
i ii 1 XY
1 n2
YYi
i 1 1
X X Y YS
cS(Y Y)
=
=
− −
= =
−
(18)
0 1c X c Y= − (19)
n
2
YY ii 1
S (Y Y)=
= − (20)
An unbiased estimator of σ2 for the regression line (17) is given in equation
(21). n
2 2EI i i
i 1
SS 1s MSE (X X )
n 2 n 2
=
= = = −− −
(21)
After the model is built, the null hypothesis H0: ci=0 versus the alternative
Ha: ci ≠ 0 is examined, as in paragraph 2.3.1, using the test statistic
i i it* (c ) s c= − .
For (1-α)100% confidence interval the null hypothesis H0: ci=0 versus the
alternative Ha: ci ≠ 0 is examined. Setting γ 1=0 the t* statistic
becomes i it* c s c= .
Given the point estimators c1 of γ1, c0 of γ0 and s2 of σ2, (1-α)100% confidence
limits for γ1 and γ0 are given by equations (22) and (23).
1 1(1 ;n 2)
2
c t s c− −
• (22)
0 0(1 ;n 2)
2
c t s c− −
• (23)
In equations (22) and (23) 2 2
1 1 YYs c s c s / S= = ,
2 2 2
0 0 YYs c s c s 1/ n Y / S = = • + .
As stated in [16], [17] (pp 392), given a new or future observation Y0, a (1-
α)100% prediction interval for the estimated value 0 0 1 0X c c Y
= + is presented
in expression (24):
0(1 ;n 2)
2
X t s pred
− −
• (24)
In (24) ( )22
I 0 YYs pred s [1 1/ n Y Y / s ]= • + + − (25)
In (25) the 2
Is given in equation (21) is used. Note that Y0 is a random variable
measured with not negligible error, so is expected to violate the fundamental
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regression assumption which states that the regressor is measured with negligible
error.
Results and Discussion
In this section a set of a calibration experiment data is analyzed with both
methods presented above and the results are compared to each other.
Data set
During the hail period of 2017, five (5) packets of Styrofoam plates have
calibrated and the calibration equations obtained for each one, using the inverse
regression method as usual. The calibration data of the fourth Styrofoam packet
will be used in the following to derive the calibration equation and a 95%
prediction band running both methods. The calibration data are included in the
following Table 2.
Table 2. Calibration data of the fourth packet of Styrofoam
X Y X Y X Y X Y X Y X Y
6.35 3.001 7.94 4.396 9.53 5.710 11.11 7.005 12.70 9.163 13.49 10.183
6.35 2.695 7.94 4.033 9.53 6.255 11.11 7.311 12.70 9.496 13.49 10.397
6.35 2.856 7.94 4.234 9.53 6.227 11.11 7.069 12.70 9.103 13.49 9.991
6.35 2.424 7.94 4.204 9.53 5.695 11.11 7.220 12.70 9.060 13.49 10.107
6.35 3.170 7.94 3.943 9.53 5.808 11.11 7.191 12.70 8.916 13.49 9.513
6.35 2.935 7.94 4.204 9.53 5.760 11.11 7.313 12.70 8.417 13.49 9.794
6.35 3.035 7.94 4.462 9.53 6.181 11.11 7.921 12.70 9.141 13.49 9.642
6.35 2.935 7.94 4.526 9.53 5.869 11.11 7.397 12.70 9.066 13.49 9.865
6.35 2.919 7.94 4.317 9.53 5.896 11.11 7.705 12.70 9.065 13.49 9.981
6.35 3.007 7.94 3.940 9.53 5.927 11.11 7.604 12.70 8.983 13.49 10.287
X Y X Y X Y X Y X Y X Y
14.29 10.754 18.26 15.825 22.23 20.543 25.40 23.597 31.75 30.462 34.93 33.443
14.29 10.583 18.26 15.870 22.23 20.137 25.40 23.805 31.75 30.186 34.93 34.083
14.29 10.397 18.26 15.715 22.23 20.290 25.40 23.758 31.75 30.381 34.93 33.268
14.29 10.494 18.26 15.713 22.23 20.151 25.40 23.572 31.75 30.480 34.93 32.941
14.29 10.178 18.26 15.698 22.23 20.458 25.40 24.207 31.75 30.995 34.93 34.378
14.29 10.061 18.26 15.485 22.23 20.913 25.40 24.490 31.75 30.583 34.93 33.951
14.29 10.716 18.26 15.579 22.23 20.796 25.40 23.883 31.75 30.669 34.93 33.885
14.29 10.597 18.26 15.488 22.23 21.163 25.40 24.375 31.75 30.252 34.93 34.053
14.29 10.502 18.26 15.316 22.23 20.670 25.40 24.294 31.75 30.565 34.93 33.978
14.29 10.620 18.26 15.538 22.23 20.493 25.40 23.936 31.75 30.308 34.93 34.130
In the above Table 2, X is the diameter of the simulation sphere expressed in
mm and Y is the minimum diameter of the dent left on the surface of the
hailpad, expressed in mm too. For each one diameter of the simulation spheres,
there are ten (10) values of the minimum dent diameter. The total number of
data pairs is n=120.
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Data analysis with the Classical Regression method
Let X to be treated as the independent (predictor) variable and Y to be
treated as the dependent (response) variable. The values Xi of X are all measured
with negligible error while the values Yi of Y are measured with random errors.
Assuming that a simple linear model is appropriate, the true model is:
i 0 1 i iY X= + + , where β0 is the Y-intercept (ordinate), β1 is the slope of the
line, and the εi’s are random errors. The random errors εi assumed to be
independent and identically distributed as normal with a mean of zero (0) and a
variance of σ2, εi~N(0,σ2). Applying the least squares method of the paragraph 2.3 with the data of the
Table 2, the regression model 0 1Y b b X
= + is obtained, where b0=-4.73033688
and b1=1.11195779 are unbiased point estimators of β0 and β1, DF=118, the
variance s2=0.2134268, the standard deviation s=0.46198138, the coefficient of
determination R2=0.99788034. In the following Table 3 there is a summary of the
values of the b0 and the b1 parameters.
Table 3. Parameters b0, b1
Parameter Value 95% confidence limits St. deviation (s) t*
b0 -4.73033688 (-4.912471921 , -4.54820184) 0.09197475 -51.4308218
b1 1.11195779 (1.102615228 , 1.121300345) 0.00471782 235.69338
For n-2=118 degrees of freedom, α=0.05 level of significance and two tailed
Hypothesis testing for the parameters b1 and b0 the t-value is
t(0.975,118)=1.98027224. The Hypothesis testing has as in the following:
Hypothesis testing for β1
Null Hypothesis H0: β1=0
Alternative Hypothesis Ha: β1≠0
Because |t*|=235.69338> t(0.975,118)=1.98027224 the Null Hypothesis is
rejected and the Alternative hypothesis is accepted, so the slope β1 is not zero,
which means that there is a statistically significant linear relation of the
minimum dent diameter to the hail diameter, at 95% level of confidence.
Hypothesis testing for β0
Null Hypothesis H0: β0=0
Alternative Hypothesis Ha: β0≠0.
Because |t*|=51.4308218> t(0.975,118)=1.98027224, the Null Hypothesis is
rejected and the Alternative Hypothesis is accepted, which states that the Y-
intercept is not zero, at 95% level of confidence.
When a new observation of a dent with minimum diameter 0Y becomes
available, from the model Y -4.73033688 1.11195779 X
= + , using the
equation (13), the mean estimated value X0 of the hail diameter is obtained. A 95% Prediction Interval for the estimated hail diameter X0 is calculated
using the equation (14). Calculations performed for minimum dent diameters
from 2mm to 35mm.
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Data analysis with the Inverse Regression method
In the case of the Inverse Regression method, the variable X, the diameter of
the simulation sphere, is treated as the response (dependent) variable and the
variable Y, the minimum diameter of the created dent, is treated as the predictor
(independent) variable. Assuming that a simple linear model is appropriate, the
true model is: '
i 0 1 iX Y = + + , where γ0 is the X-intercept (ordinate), γ1 is
the slope of the line and '
i are random errors. The random errors '
i assumed to
be independent and identically distributed as normal with a mean of zero (0) and
a variance of σ2, '
i ~N(0,σ2).
Applying the Least Squares method of the paragraph 2.3 on the data of the
Table 2, the regression model 0 1X c c Y
= + is obtained, where c0=4.28176748
and c1=0.89740848 are unbiased point estimators of γ0 and γ1 respectively,
DF=118, variance s2=0.17224666, standard deviation s=0.4150261 and the
coefficient of determination R2=0.99788034. In the following Table 4 there is a
summary of the values of the c0 and the c1 parameters.
Table 4. Values of the parameters c0 and c1
Parameter Value 95% confidence limits St. deviation (s) t*
c0 4.28176748 (4.148957984 , 4.414576973) 0.06706628 63.8438185
c1 0.89740848 (0.88986854 , 0.904948412) 0.00380753 235.69338
For n-2=118 degrees of freedom, α=0.05 level of significance and two tailed
testing t(0.975,118)=1.98027224, the Hypothesis testing has as in the following:
Hypothesis testing for γ1
Null Hypothesis H0: γ1=0
Alternative Hypothesis Ha: γ1≠0
Because |t*|=235.69338> t(0.975,118)=1.98027224 the Null Hypothesis is rejected
and the Alternative Hypothesis is accepted, so the slope γ1 is not zero, which
means that there is a statistically significant linear relation of the hail diameter
to the minimum dent diameter, at 95% level of confidence.
Hypothesis testing for γ0
Null Hypothesis H0: γ0=0
Alternative Hypothesis Ha: γ0≠0
Because |t*|=63.8438185> t(0.975,118)=1.98027224, the Null Hypothesis is
rejected and the Alternative Hypothesis is accepted, which states that the X-
intercept (ordinate) γ0 is not zero, at 95% level of confidence.
When a new observation of a dent with minimum diameter Y0 becomes
available, the mean estimated value X0 of the hail diameter is obtained based on
the model X 4.28176748 0.89740848 Y
= + and a 95% Prediction Interval
for the hail diameter X0 is calculated using the equations (24) and (25).
Calculations performed for minimum dent diameters from 2mm to 35mm.
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Comparison of the results of the two models and discussion
In the following Table 5, the prediction values and the 95% prediction
intervals of the calculations performed in the paragraphs 3.2 and 3.3 are
presented.
Table 5. Comparison of the results of the Classical and Inverse regression methods
Yo Xo_cl CLl CLup Xo_In Inl Inup ΔXo(Cl-In) ΔXo_l ΔXo_up
2 6.053 5.221 6.884 6.077 5.246 6.907 -0.024 -0.025 -0.023
3 6.952 6.121 7.783 6.974 6.144 7.804 -0.022 -0.023 -0.021
4 7.851 7.021 8.681 7.871 7.042 8.700 -0.020 -0.021 -0.019
5 8.751 7.921 9.580 8.769 7.940 9.597 -0.018 -0.019 -0.017
6 9.650 8.821 10.479 9.666 8.838 10.494 -0.016 -0.017 -0.015
7 10.549 9.721 11.377 10.564 9.736 11.391 -0.014 -0.015 -0.013
8 11.449 10.621 12.276 11.461 10.634 12.288 -0.012 -0.013 -0.012
9 12.348 11.521 13.175 12.358 11.532 13.185 -0.011 -0.011 -0.010
10 13.247 12.420 14.074 13.256 12.430 14.082 -0.009 -0.010 -0.008
11 14.147 13.320 14.973 14.153 13.328 14.979 -0.007 -0.008 -0.006
12 15.046 14.219 15.872 15.051 14.225 15.876 -0.005 -0.006 -0.004
13 15.945 15.119 16.771 15.948 15.123 16.773 -0.003 -0.004 -0.002
14 16.844 16.018 17.671 16.845 16.020 17.671 -0.001 -0.002 0.000
15 17.744 16.918 18.570 17.743 16.918 18.568 0.001 0.000 0.002
16 18.643 17.817 19.469 18.640 17.815 19.466 0.003 0.002 0.004
17 19.542 18.716 20.369 19.538 18.712 20.363 0.005 0.004 0.006
18 20.442 19.615 21.268 20.435 19.609 21.261 0.007 0.006 0.007
19 21.341 20.514 22.168 21.333 20.507 22.158 0.009 0.008 0.009
20 22.240 21.413 23.068 22.230 21.404 23.056 0.010 0.010 0.011
21 23.140 22.312 23.967 23.127 22.301 23.954 0.012 0.011 0.013
22 24.039 23.211 24.867 24.025 23.198 24.852 0.014 0.013 0.015
23 24.938 24.110 25.767 24.922 24.094 25.750 0.016 0.015 0.017
24 25.838 25.008 26.667 25.820 24.991 26.648 0.018 0.017 0.019
25 26.737 25.907 27.567 26.717 25.888 27.546 0.020 0.019 0.021
26 27.636 26.806 28.467 27.614 26.785 28.444 0.022 0.021 0.023
27 28.536 27.704 29.367 28.512 27.681 29.342 0.024 0.023 0.025
28 29.435 28.602 30.267 29.409 28.578 30.241 0.026 0.025 0.027
29 30.334 29.501 31.168 30.307 29.474 31.139 0.028 0.027 0.028
30 31.234 30.399 32.068 31.204 30.371 32.038 0.029 0.029 0.030
31 32.133 31.297 32.968 32.101 31.267 32.936 0.031 0.030 0.032
32 33.032 32.196 33.869 32.999 32.163 33.835 0.033 0.032 0.034
33 33.931 33.094 34.769 33.896 33.059 34.733 0.035 0.034 0.036
34 34.831 33.992 35.670 34.794 33.955 35.632 0.037 0.036 0.038
35 35.730 34.890 36.571 35.691 34.851 36.531 0.039 0.038 0.040
In the Table 5, given a new observation Y0, the mean predicted value of the
hail diameter using the classical regression method is Xo_cl, the lower prediction
limit is CLl and the upper prediction limit is CLup, and the mean predicted value
of the hail diameter using the inverse regression method is Xo_In, the lower
prediction limit is Inl and the upper prediction limit is Inup, ΔXo(Cl-In) is the
difference of the predicted values between the classical and the inverse method,
Open Science Journal Research Article
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and ΔXo_l and ΔXo_up are the differences between the two methods for the lower
and upper prediction limits respectively.
Looking at the values of the Table 5 it can be seen that the differences
between the mean estimated values X0 of the hail diameter with the two methods
are very small and any difference appears in the second decimal. Given that the
unit of measurement is the millimetre (mm), the calculated differences of the
Table 5 in the predicted values with the two regression methods can be ignored,
compared to the errors coming from the large variability of the natural hail [9]
and other sources of errors [12].
It can also be seen in the Table 5 that for values of Y0 from 2mm to up to
14mm the average estimated value X0 with the Classical regression method is less
than that with the Inverse regression method, while for values greater than
15mm the estimated values with the Classical method are larger than that
estimated with the Inverse method. The lower and upper limits of 95% confidence
intervals show the same behaviour. The slightly greater slope of the regression
line of the Classical method is expected mathematically and does not change the
general conclusion that the results are the same.
Concluding remarks and recommendations
In this paper two statistical methods for the calibration of the hailpads
examined, calibration using the Classic Regression method and calibration using
the Inverse Regression method. Some principles of statistical inference were also
applied to examine the extent to which the results are statistically acceptable.
Finally, the two methods were applied to the data of a calibration experiment
taken using the Energy Matching technique and the results were compared to
each other. The comparison shows that the results are similar.
While the Classic Regression method is preferable from a statistical point of
view, since it takes the diameter of the hail as the independent variable, the
Inverse Regression method gives almost the same results and it is easier, not so
much to obtain the calibration equation but mainly for the calculation of the
prediction bands. It is therefore not advisable to replace the Inverse Regression
method used so far in the GNHSP with the Classical Regression one, but it is
recommended both methods to be used, along with the introduction of the
statistical inference procedures presented in this work.
The analysis of the hit hailpads of the Greek network of the years 2008 to
2017 shows that the diameter of the hail rarely exceeds 26mm, so it is
recommended the calibration range to extend till this limit and the calibration
experiment to performed using a new set of calibration steel spheres having
diameters of the metric system from 6mm to 26mm, increasing by two (2) mm. If
some rare hail fall will occur with diameters of hailstones greater than 26mm,
then a special calibration can be performed, using calibration spheres with
diameters greater than 26mm. In such a case second order polynomials can also
be tried for better fitting of the model.
Acknowledgements
I would like to thank Dr José Luis Sanchez, Professor of Applied Physics,
University of León, Spain, for the hospitality he offered me at his Laboratory and
Open Science Journal Research Article
Open Science Journal – October 2019 16
the fruitful discussions with him and his collaborators about the calibration of the
hailpads, especially in the topic of the good laboratory practices to minimize the
error sources, which helped me to proceed to some improvements in the
calibration procedures at the hailpad laboratory of ELGA. I would like to
acknowledge my collaborators Mr. Tegoulias Ioannis, Physicist – Meteorologist,
for his involvement in the calibration procedure, and Mr. Amarantides
Konstantinos, hailpad technician, who carries out a large part of the calibration
work. In conclusion, I would like to thank the anonymous reviewer of the present
work, whose helpful comments and suggestions were valuable to me.
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